T.W.I.G.S. The "What Is ...?" Graduate Seminar Room 1634, Lederle Graduate Research Tower University of Massachusetts, Amherst
The Seminar meets on most Fridays, 2:45-3:45 in 1634 LGRT.     Driving Directions   and   Campus Maps Overview of the Seminar

		26 September	Farshid Hajir, UMass Amherst
		2:45-3:45	What Is A Newton Polygon?
		3 October	Eric Sommers, UMass Amherst
		2:45-3:45	What Is A Root System?
		17 October	George Avrunin, UMass Amherst
		2:45-3:45	What Is A Finite-State Verification?
		24 October	Tom Braden, UMass Amherst
		2:45-3:45	What (else) Is A Newton Polytope?
		7 November	Farshid Hajir, UMass Amherst
		2:45-3:45	What Is an L-Function?
		14 November	Tom Weston, Amherst College
		2:45-3:45	What Is A Motive?
		21 November	Walter Rosenkrantz, UMass Amherst (tentative)
		2:45-3:45	What Is The Black-Scholes Options Pricing Formula?
		5 December	Panos Kevrekidis, UMass Amherst
		2:45-3:45	What Is A Blowup?
		12 December	Paul Gunnells , UMass Amherst
		2:45-3:45	What Is A Modular Form?

Abstract
To a polynomial with rational coefficients and a prime p, we attach in a simple way a polygon in the plane, its p-adic Newton Polygon, which captures a surprisingly large amount of data concerning the roots of f. I'll describe the origins of this concept in the work of Newton, its development in the hands of Hensel and Dumas, and its applications to algebraic questions such as factorization and calculation of Galois groups.

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A root system is a collection of vectors in a Euclidean space satisfying certain axioms. I will present these axioms, describe the classification of root systems, and explain some uses of root systems in Lie theory. I'll finish with a discussion of some current problems that require nothing more than the definition of a root system.
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Finite-state verification is a collection of techniques that attempt to answer the questions "Does a given computer system always do what it is supposed to do?" and "Does it ever do what it's not supposed to do?" Anyone who has used a large computer program knows that the easy answers are almost certainly "no" and "yes", respectively; the hard part is determining exactly how and under what circumstances the system may misbehave. Finite-state verification techniques work by constructing finite models that represent all the executions of the system, and then apply various algorithms to check specific properties of the system. I'll describe some of these techniques and the interesting mathematical issues that arise in trying to make these techniques practical for real systems.
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It has been known for a long time that "most" polynomials of degree $n$ in one variable have $n$ distinct solutions in $\mathbb C$. One can ask the same question for a system of $d$ polynomials in $d$ variables. The notion of "degree" is replaced by the Newton polytope, a convex region which controls what monomials are allowed. The formula for the number of solutions is given by the "mixed volume" of the Newton polytopes, a classical quantity from convex geometry studied by Minkowski. If time permits, we will also discuss another generalization involving a single polynomial in several variables; since there will be infinitely many solutions, the question must be rephrased in terms of the "Euler characteristic", a topological invariant which generalizes the size of a finite set of points. TOP

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An L-function is a way to package an arithmetic function (a sequence of numbers) into an analytic object, whose analytic properties (special values, residues, poles, analytic continuation, functional equation, order of growth, ...) give the most revealing portrait of the arithmetic data. The grand-daddy of all L-functions is the Riemann Zeta Function. I will concentrate on the case of elliptic curves, illustrating how L-functions act as an analytic bridge between arithmetic/algebra and geometry.
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Roughly speaking, a motive is an arithmetic object coming from geometry. Attempts to make that idea precise have frustrated mathematicians for the forty years since Grothendieck initiated the subject. In this talk we will focus on the reasonably concrete example of the motive of an elliptic curve and its symmetric square. We will also hope to give some idea of how the formalism of motives can be used to investigate arithmetic questions.
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What is a fair price to pay for a financial asset with an uncertain future return? We are particularly interested in determining the fair price of an option, which is the right to buy (call option), or sell (put option), a financial asset, e.g. stocks or bonds, at a fixed price K called the strike price, by a certain date, called the expiration date. The Black-Scholes options pricing formula solves this problem via the concept of a ``replicating portfolio". The mathematical derivation of this result is greatly simplified by first solving the problem in the context of the ``single period binomial lattice Model". A straight forward induction argument yields the options pricing formula for the multi-period case, and a suitable application of the central limit theorem yields the general result. In the course of deriving the Black-Scholes options pricing formula we discuss the ``efficient markets hypothesis", the random walk model for stock price fluctuations, and related questions of independent interest.
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In this talk we'll discuss self-similar blowup ("focusing") of waves in physical systems. As a benchmark example of a PDE whose solutions display such behaviour, we will examine the nonlinear Schr{\"o}dinger equation (NLS). We 'll review the main wave solutions of the equation and their spectral properties, revealing how they give rise to blowup as the relevant parameters are varied. This effect will then be examined through a (very general) dynamic renormalization point of view that allows one to observe and justify the emergence of focusing solutions. The implications of such a renormalization viewpoint on stability, the normal form of this phenomenon, analogies of the self-similar blowup solutions with travelling wave ones, and generalizations/extensions of this framework will also be briefly discussed.
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Last modified: 03 September 2003 by Farshid Hajir